Modulators have been used to modulate the phase of an optical wave in an optical fiber. These modulators have used acoustic waves to periodically compress and decompress the fiber from an unperturbed value to periodically increase and decrease the fiber's density. This causes the refractive index of the fiber to periodically vary. When the acoustic wave is large in wavelength compared to the area of the beam, then the light beam interacts with a region that will appear to have a uniformly varying index of refraction. The optical thickness of the region is the physical thickness times the index of refraction. If the index of refraction varies sinusoidally, then the optical thickness does too, and the phase of the light signal, when measured, will have a sinusoidal perturbation proportional to the change in index of refraction, and thus to the acoustic wave that caused the perturbation.
If an acoustic strain wave of amplitude S travels through a region with unperturbed index of refraction `n`, then the change in `n` will roughly be given by .DELTA.(1/n.sup.2).apprxeq.B, where B is known as the relative impermeability, and .DELTA.B.apprxeq.p*S where S is the acoustic strain and p is a strain-optic coefficient. The strain optic coefficient is (a number determined by the material properties of the substance in which the optical and acoustic waves are interacting. In full form, this is expressed as a tensor, or matrix, equation that takes into account that the index of refraction, n, in a material, is not necessarily the same in all directions. A good example of a material with this non-isotropic nature is natural calcite. The analysis below can be extended to a full three dimensional tensor analysis with arbitrary materials and propagation angles of the optical and acoustic waves. No essential generality is lost in the following pseudo- one-dimensional analysis, since it is merely a component analysis of the full problem that can be added to essentially identical solutions for the other tensor components.
When the acoustic wave is at an angle to the optical wave, then the acoustic wave can interact with the light wave, changing the phase and polarization of the light wave, and this interaction will depend upon the relative and, sometimes, the absolute directions of the acoustic and optical beams.
The next important concept is that an acoustic wave can cause three distinct effects. Imagine a system with an optical signal traveling in the Z direction (out of the paper) and zero in the Y direction. Imagine then that it has polarization with some component in the X direction and some component in the Y direction. Now let an acoustic wave in the X-Y plane interact with the light wave. First, imagine the acoustic wave is also traveling in the X direction. It can change the effective density in the X direction and thus couple from X to X where the net effect is to alter the phase of the X component of the optical beam. Second, if you are compressing the material X, it will, by analogy, bulge outward in Y. Thus the signal in Y is also affected, albeit to a different degree.
Third, and most pertinent to the modulator, in addition to changing the phase within the X and Y polarizations, it can also couple some of the signal from the X polarization into the Y polarization, and vice versa. Thus, the acoustic wave can be used to couple between polarizations in an optical signal. This will be referred to later as either cross coupling or the .DELTA.B.sub.6 term where .DELTA.B.sub.6 is a term in the impermeability tensor.
A very strong and familiar analogy is the use of standard optical polarizers with an optical signal. Without loss of generality, imagine a polarization plate with polarization axis (polarization direction at which light passing through the polarizer will be unaffected) in the X direction. If an optical signal with Amplitude A strikes the polarizer with the signal polarization at some angle .theta. to the X direction, then the light that emerges through the plate will have amplitude Acos(.theta.) and be polarized in the X direction. Most importantly, note that 100% of the signal is passed if .theta.=0, passed if .theta.=90.degree..
Therefore, if a second polarizer is placed after the first, with polarization axis at an angle .phi. to X, then the process is iterated since the beam emerging from polarizer #1 is polarized in the X direction. The beam will emerge from the second polarizer reduced in amplitude by cos(.phi.) and polarized at angle .phi. from the X direction. Again, if this second polarizer is aligned in the Y direction, then zero signal will pass through since, with X and Y perpendicular,.phi.=90.degree. and cos(.phi.)=.phi..
The cross coupling effect, mentioned above, will have the same general effect as putting a third polarization plate in between the two crossed ones. If the third plate is parallel to either of the others, then nothing is changed. Say, however, that it is placed at an intermediate angle, say some angle .theta..pi.away from X, and that the signal emerging from polarizer #1 has amplitude A in the X direction. Then the signal will pass through the intermediate polarizer with amplitude Acos(.theta.), polarized at an angle .theta. from X. It will then hit the Y oriented polarizer at an angle (90.degree.-.theta.) away from the Y oriented axis (since X and Y are 90.degree. apart). Thus the input signal will be further reduced from amplitude Acos(.theta.) to [Acos(.theta.)]cos(90.degree.-.theta.) which is the same as Acos(e)sin(e). Thus, excepting .theta.=0 or 90 degrees, a non-zero component will be coupled across. The effect is most efficient when .theta.=45.degree..
The shortfall of such a system is that it will be, at best, very. inefficient, and, at worst, utterly ineffectual if the angle between the acoustic wave and optical polarization is 0 or 90.degree.. The effect is basically saying that the two polarization modes, or states of the optical signal are usually orthogonal, or, in other words, that there would normally be no coupling between them. The introduction of the acoustic wave is like placing an intermediate polarizer in between these orthogonal ,modes so that coupling can take place. It depends, however, on this intermediate angle being picked and maintained optimally.
Where the plane of .polarization of the optical wave in the fiber is at an angle with respect to the direction of compression and expansion, this has resulted in modulation of the plane of polarization of the optical wave traveling in the fiber. The modulation is then detected by an optical analyzer followed by an optical detector which receives the optical wave and provides an output electrical signal representative of the modulation. Problems arise because of changes in the plane of polarization along the optical fiber due to changes in temperatures and position of the optical fiber and other environmental conditions to which the optical fiber is subjected.
One type of modulator excites and propagates longitudinal acoustic waves across the fiber in the x direction. The periodic compression and expansion of the fiber due to the acoustic waves causes the polarization state of a transmitted optical wave to shift. A specific example is an optical wave polarized at 45.degree. to the x-direction to have its polarization shifted to 135.degree. with respect to the x-direction. If an analyzer is placed at the output of the optical transmitting fiber and its output applied to a detector, an amplitude modulated signal will be generated by the transducer, whose frequency is dependent on the frequency of the electrical drive signal. A signal can be transmitted by varying the drive power to the modulator, in the extreme, by turning the modulator on and off, responsive to the signal. As described above, the polarization state (circular or linear) can easily change in a long fiber communication system. There could be a zero output response if correct angles of excitation and detection are not maintained.